Soret and Dufour effects on unsteady MHD second-grade nanofluid flow across an exponentially stretching surface

The unsteady energy and mass transport of magnetohydrodynamics (MHD) second grade nanofluid via an exponentially extending surface with Dufour and Soret effects are investigated in this study. Variable thermal conductivity and mixed convection effects are used to investigate the heat transfer mechanism. There are also new characteristics such as slip flow, viscous dissipation, Brownian motion, nonlinear thermal radiation, and thermophoresis. In the problem formulation, the boundary-layer approximation is used. Using the suitable transformations, the energy, momentum, and concentration equations are generated into non-linear ordinary differential equations (ODEs). The solution to the resultant problems was calculated via the Homotopy analysis method (HAM). The effects of environmental parameters on velocity, temperature, and concentration profiles are graphically depicted. When comparing the current results to the previous literature, there was also a satisfactory level of agreement. In comparison to a flow based on constant characteristics, the flow with variable thermal conductivity is shown to be significantly different and realistic. The temperature of the fluid grew in direct proportion to the thermophoresis motion, buoyancy ratio, and Brownian motion parameters. According to the findings, the slippery porous surface may be employed efficiently in chemical and mechanical sectors that deal with a variety of very viscous flows.


Mathematical model
The time-dependent, 2D incompressible flow of MHD viscoelastic (Second-grade) nanofluid induced by an exponentially stretching surface is taken into account in this study, as exposed in Fig. 1. u w (x, t) = ae x / L /(1 − ct) represents the surface is stretched with the exponential velocity, where v o is constant, L indicates the characteristic length, and c denotes the unsteadiness. The ambient and reference temperatures and concentrations are designated as T w , T ∞ , C w , and C ∞ respectively, whereas T w = T ∞ + e x / 2L (T o /(1 − ct)) and C w = C ∞ + e x / 2L (C o /(1 − ct)) determines the temperature and concentration distribution near the surface. The magnetic field is assumed to be B(t) = B o /(1 − ct) 0.5 e x / 2L , with B o denoting a uniform magnetic field. The characteristics of the physical model and the mathematical model are considered under the following environments: • Viscous dissipation.
• Impacts of Brownian and thermophoresis motion.
According to the following assumptions, the governing equations for continuity, momentum, heat and constration when using the BL approximation are 12,[55][56][57][58][59][60]64 : the boundary conditions are:  www.nature.com/scientificreports/ here v and u indicates the velocity component along the y -axis, and x -axis respectively, whereas the fluid temperature T. ρ f the density, gravity is g, The dynamic viscosity is µ f , δ f the electrical conductivity, Stefan-Boltzmann constant δ f , mean absorption constant k * , Brownian movement coefficient D B , ρCp f is the heat capacity, magnetic field strength B o and D T thermophoresis diffusion coefficient.
In the temperature range of 0-400 F, thermal conductivity varies linearly. Variable thermal conductivity is therefore approximated as 53 : where the empirical constants δ * , K(T) variable thermal/heat conductivity, and k * coefficient of heat conductivity are far from the exponential plate. For fluids, 0 < δ * < 0 can be positive, but for gases, it can be negative.
The following similarity transformations are illustrated for the governing Eqs. (1)-(4) with the constraints (5) in a much simple way 59 . Where the stream function ω can be specified as v = −∂ω ∂x, and u = ∂ω ∂y, while the similarity variable is η : Equations (2)-(5) may be reduced to set of nonlinear ODEs in the setting of the above-mentioned relations by using the similarity transformations (6): with the constraints where the unsteadiness parameter is β = 2Lc /ae x/L , the magnetic parameter is The coefficient of skin friction C fx , the local Nusselt number (Nu x ) and the local Sherwood-number (Sh x ) are thus defined as: www.nature.com/scientificreports/ Then we employ (6) into (11), (12) and (13), yielding the following relationship: where Re x = u e x ν f is the x-axis local Reynolds number.

Solution by HAM.
For solving highly nonlinear equations, the HAM is a powerful analytical tool. In light of boundary conditions 10, the HAM is used to resolve the generated Eqs. (7)(8)(9)(10). The linear operators and Initial guesses are required to start the procedure using this method. As a result, we used � f , � θ , � φ as linear operators and f 0 (η), θ 0 (η), φ 0 (η) as initial estimations to solve momentum, energy and mass transform equations using the aforesaid method. For additional information on this technique, see 9-12,56-63 .
The properties of the aforesaid operator are as follows: where C js ( j = 1, 2, …7) are arbitrary constants.
where h f , h φ and h θ signify non-zero auxiliary parameters and q ∈ [0, 1] represents an embedding parameter while F ,θ andφ representing the mapping occupations for f , θ and φ. The boundary conditions become

Results and discussion
On a second-grade nanofluid stream over an exponentially surface, an analysis of unsteady flow was performed.
Due to the combination of viscous and nonlinear radiative heat, an electrically charged fluid is investigated in this scenario. An approximate analytical approach called HAM is used to handle the altered equations created from the said model. Examine the effects of dynamic parameters of the contours of our system on velocity, and temperature. Also, the comparison of variable and constant thermal conductivity through graphs. We have decided on the settings of key parameters for our simulation as M = 0. To verify the numerical values of θ ′ (0) with Haider et al. 64 , Table 1 was created. The numerical values acquired by the HAM in the current investigation were shown to be in great agreement with the literature.
The effect of the magnetic parameter (M) on flow rate appears in Fig. 2. When M is elevated, the flow rate upsurges as well. Physically, magnetic strength provides resistive forces called Lorentz forces that oppose fluid flow, therefore, the fluid velocity drops. The deviation of the second-grade parameter ( α ) on motion is portrayed in Fig. 3. It is revealed that a rise in α leads to an enhancement in velocity of both liquid and hybrid nanofluid. The reason is that, as alpha increases the viscous forces and fluid viscosity fall. Figure 4a,b demonstrates the       Figure 5 examines the influence of an unsteady component ( β ) on motion (a) and heat (b) profiles. With increasing β, the temperature and velocity profiles drop. This is because raising β decreases the momentum and thermal BL. The relationship between the radiation parameter Nr and the temperature ratio parameter (θ w ) with the liquid temperature is seen in Fig. 6a,b. The random motion of particles is assisted by higher estimates of the Nr. Consequently, more particle collisions are observed, and more heat is created. So, there is a rise in the fluid heat. The thermal profile grows when the temperature ratio parameter (θ w ) upsurges, as perceived in Fig. 6b. These results are indicated that when θ w grows, the temperature difference (T w − T 0 ) rises, causing the fluid temperature to rise. Figure 7 demonstrates that the behaviour of Ec is directly proportional to the temperature field, as Ec rises viscous dissipation upsurges, i.e., Heat energy is converted from kinetic energy, which raises the temperature. The inspiration of the thermophoresis parameter (Nt) on the heat flux is exposed in Fig. 8a. For larger values of Nt, the heat transfer rises. Thermophoresis is a transportation force that arises when a temperature differential exists between fluid layers. With a higher Thermophoresis value, the temperature differential between the layers grows, and the heat transformation rate grows as well. As a result, the temperature progressively rises, increasing the kinetic energy of nanoparticles. Figure 8b describes the inspiration of the  www.nature.com/scientificreports/ Brownian motion parameter (Nb) on the heat flux. A rise in the Nb caused the temperature profile to settle at higher levels. Nb is the random motion caused by nanoparticles colliding with the base fluid. The Nb increases, and the collision rises. The internal kinetic energy of the fluid increases due to particle collisions. Figure 9a demonstrates the upshot of the Dufour (Df) on the temperature. Du denotes the caloric energy discharge during the flow being influenced by attention gradients. This is shown when the Du increases and the temperature distribution rises monotonically. The inspiration of the thermal conductivity parameter (δ) on heat flux is seen in Fig. 9b. The graph shows that when δ is amplified, the temperature escalations. Low thermal conductivity fluids have low temperatures, whereas high thermal conductivity fluids have high temperatures, as is well known. Larger thermal conductivity indicates that kinetic energy among molecules is more heightened, due to a great number of molecular collisions. The kinetic energy is transformed into thermal energy more quickly, resulting in more heat transfer. Figure 10a exhibitions the impact of Soret number (Sr) on the concentration profile. The temperature difference to the concentration quotient is denoted by Sr. The temperature gradient increases as Sr increases. The rate of molecular diffusion is thought to be increasing. As a result, for increasing Sr levels, the rate of mass transfer accelerates. Therefore, φ(η) improves. Figure 10b shows the upshot of the Schmidt number (Sc) on the particle concentration. With growing Sc, the concentration profile drop. The Sc is calculated by dividing the viscous rate of diffusion by the molecular rate of diffusion. Because of stabilising the molecular diffusion rate, the viscous diffusion rate increases, raising the Sc.
As seen in Fig. 11, the f ′′ (0) growth as M increases while decreasing as α rises. Increasing the value of M causes a significant resistance to fluid flow owing to the Lorentz drag force, which decreases the fluid velocity and the momentum BL thickness, increasing the velocity and, as a result, the shear stress at the exponential stretching   Fig. 13. The reason for this is that larger values of Nr and σ cause nanoparticles to move away from the surface and into the free stream, generating friction between the surface and fluid and slowing heat transmission. The fluctuations of −φ ′ (0) with Sc for various amounts of Sr are shown in Fig. 14. According to the graph, the mass transfer rates rise as Sc grows, with slightly different behaviour for Sr. In the presence of a lower Sc force generated by a temperature gradient, the intensity of random movement of nanoparticles boosts their transportation rate, whereas, in the presence of a larger magnitude of Sr force, the transport rate declines.

Conclusions
In this article, we have examined the time-dependent slip boundary layer flow of second-grade nanofluid over an exponentially stretching surface under the variable thermal conductivity using HAM. Also, The Soret, Dufour, and nonlinear thermal radiation impacts were all taken into account. The validity of major findings and outcomes was discussed. When the results were compared to the existing literature, they were found to be good. The impact of the relevant parameters on flow, heat, and mass transmission characteristics via a graph and briefly discussed. Therefore, some final remarks of this article are as follows.    the thermal radiation, temperature ratio, Eckert number, Brownian motion, variable thermal conductivity parameter and thermophoresis have a substantial influence on heat transfer. • Dufour and Soret numbers have a significant impact on the quicker heat and mass transport in variable thermal conductivity compared to constant thermal conductivity respectively. • An enhancement in the thermal conductivity parameter indicates a temperature upsurge.
• The local Nusselt number declined by growing variable conductivity and thermal radiation parameters.
• The skin friction coefficient improves with magnetic fields, Brownian motion, and thermophoresis, but diminishes when the value of a second-grade parameter increases. • The local Sherwood number decline by growing the Soret number, but rise when the value of a Schmidt number increases. • When compared to the assumption of constant fluid characteristics, the variable thermal conductivity is noticeably different.

Data availability
The data used to support the findings of this study is available from the corresponding author upon request.